Another estimator referred to as noise-contrastive estimator (InfoNCE) [1]} is defined as
\(I(X;Y) & \ge I_{NCE}(X;Y) \nonumber \\& = \mathbb {E}_{(\mathbf {x},\mathbf {y})\sim p_{XY,N}(\mathbf {x},\mathbf {y})}\biggl [ \frac{1}{N} \sum _{i=1}^{N}{ \log \biggl ( \frac{e^{T_{\theta }(\mathbf {x_i},\mathbf {y_i})}}{\frac{1}{N} \sum _{j=1}^{N}{e^{T_{\theta }(\mathbf {x_i},\mathbf {y_j})}}}\biggr )} \biggr ],\) 
